Full Text - Journal of Applied Mathematics and Computational

Journal of Applied Mathematics and Computational Mechanics 2014, 13(1), 39-44
FREE VIBRATION OF AXIALLY FUNCTIONALLY GRADED
EULER-BERNOULLI BEAMS
Stanisław Kukla, Jowita Rychlewska
Institute of Mathematics, Czestochowa University of Technology
Częstochowa, Poland
[email protected], [email protected]
Abstract. In this contribution, free vibration of axially functionally graded beams is analysed within the framework of the Euler-Bernoulli beam theory. The beams with uniaxial
variation of the elasticity modulus and mass density are approximated by an equivalent
beam with piecewise exponentially varying geometrical and material properties. A numerical example for a beam with pinned ends is presented.
Keywords: axially functionally graded beams, free vibration, Euler-Bernoulli beam theory
Introduction
Functionally graded (FG) beams are composites characterized by the volume
fraction of different materials (ceramic and metal) which is varied continuously
with the thickness and/or the length of the beam. Through an appropriate selection
of the volume fraction the FG beam with expected thermal and mechanical properties can be obtained. Therefore, the FG beams can be used in various engineering
applications.
Vibration of non-homogenous beams is the subject of investigations presented
in many papers. The free vibrations of FG beams was analyzed by using various
methods in papers [1-5]. The finite element method to the free vibration problem of
a FG beams was applied by Alshorbagy et al. in paper [1]. The presented analysis
concerns the FG beams assuming a simple power law of a change of the material
distribution through the thickness or the longitudinal direction. Hein and Feklistova
[2] present an application of the Haar wavelet approach to free vibrations of FG
beams with various boundary conditions and varying cross-sections. In paper [3]
by Anandrao et al. the finite element system of equations to free vibration analysis
of the FG beams is derived. The variation of material properties across the thickness of the beam was governed by a power law distribution. The free vibration and
stability of tapered beams made of axially FG materials were studied by Shahba
and Rajasekaran [4]. The solution to the problem was obtained by applying a differential transform element method. The exact solution to free vibration of axially
exponentially graded beams is presented by Li et al. in reference [5]. Free vibration
40
S. Kukla, J. Rychlewska
of axially FG beams consisting of two segments was studied in paper [6]. The analytical solution to the problem was obtained by assumption that the changes of the
cross-sectional area and material properties in the beam segments have an exponential form.
This paper presents a solution to the problem of free vibration of a beam consisting of an arbitrary number of axially exponentially graded beam segments.
The frequency equation is numerically solved. The eigenfrequencies are presented
in a tabular form.
1. Mathematical formulation of the problem
Consider an axially graded and non-uniform beam of length L. It is assumed
that material properties and/or cross-section of the beam vary continuously along
the length direction. According to the Euler-Bernoulli beam theory, the governing
differential equation is given by
∂2 
∂ 2w 
∂2w
(
)
(
)
(
)
(
)
E
x
I
x
+
ρ
x
A
x
= 0,


∂x 2 
∂x 2 
∂t 2
0 < x < L,
(1)
where x is axial coordinate, w( x, t ) is the transverse deflection of the beam at the
position x and time t, E (x ) is the Young’s modulus, I ( x ) denotes the moment of
inertia, ρ ( x ) and A( x ) denote the mass density and cross-sectional area, respectively. In this contribution it is assumed that
E ( x )I ( x ) = Df (x )
ρ ( x ) A(x ) = mf ( x ),
0 < x < L,
(2)
where D is a reference value of EI at x = 0 , m is a reference value of ρA at x = 0
and f is a function which has continuous derivatives up to the second order.
The function f (x ) in intervals xi −1 < x < xi , i = 1,.., n , is approximated by the functions of exponential form, i.e.
f (x ) ≅ di e
2 βi
x
L
for xi −1 < x < xi ,
with x0 = 0 and xn = L . It is assumed that f (xi ) = di
account equations (3) in equations (2), one obtains
E ( x )I ( x ) ≅ Dd i e
2 βi
x
L
2 βi
x
L
ρ ( x )A( x ) ≅ md i e
i = 1,..., n
x
2 βi i
e L
for xi −1 < x < xi ,
(3)
, i = 0,1,.., n . Taking into
i = 1,..., n
(4)
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Free vibration of axially functionally graded Euler-Bernoulli beams
Denoting the length of the i-th element as ∆xi = xi − xi −1 , the gradient parameters
βi are given by
βi =
L
f ( xi )
ln
,
2∆xi
f ( xi −1 )
(5)
i = 1,..., n
Assuming the transverse displacement of the beam in the form w(x, t ) = wi (x, t ) for
xi −1 < x < xi , i = 1,..., n , the governing equation for such a piecewise beam can be
expressed by
∂2
∂x 2
x
x

2 βi
2βi
∂ 2 wi 
∂ 2 wi
L
L
Dd
e
+
md
e
= 0,
 i
i
2 
∂x 
∂t 2

xi −1 < x < xi ,
i = 1,..., n
(6)
Setting
wi (x, t ) = Wi (x )sin ω t ,
(7)
i = 1,..., n,
where Wi ( x ) , i = 1,..., n , are the corresponding amplitude functions and ω is the
circular frequency of the beam, one obtains
x
x
2 βi
d 2  2 βi L d 2Wi 
2
L
De
−
m
ω
e
Wi = 0,


dx 2 
dx 2 
xi −1 < x < xi ,
i = 1,..., n
(8)
Introducing the non-dimensional variables
ξ=
x
,
L
Ω4 =
m 4 2
Lω
D
(9)
equations (8) can be rewritten as
d2
dξ 2
 2 βiξ d 2Wi 
− Ω 4e 2 βiξ Wi = 0,
e
2 
d
ξ


ξ i −1 < ξ < ξ i ,
i = 1,..., n
(10)
After some transformations we can rewrite equations (10) as follows:
d 4Wi
d 3Wi
d 2Wi
+ 4 βi
+ 4 βi2
− Ω 4Wi = 0,
4
3
2
dξ
dξ
dξ
ξ i −1 < ξ < ξ i ,
i = 1,..., n (11)
2. Solution of the free vibration problem
The general solution of equations (11) has the following form
Wi (ξ ) = e − β i ξ (Ai cos δ iξ + Bi sin δ iξ + Ci cosh δ iξ + Di sinh δ iξ ), ξ i −1 < ξ < ξ i , (12)
where δ i = Ω 2 − βi2 , δ i = Ω 2 + β i2 , i = 1,..., n .
42
S. Kukla, J. Rychlewska
The above solutions will be applied to boundary conditions for pinned-pinned
beam as follows
W1 (0 ) = Wn (1) = 0
d 2W1
d 2Wn
(
0
)
=
(1) = 0
dξ 2
dξ 2
(13)
The matching conditions between two connecting elements of the piecewise beams
satisfy the following continuity conditions
Wi (ξ i ) = Wi +1 (ξ i )
d 2Wi
d 2Wi +1
(
)
(ξ i )
ξ
=
i
dξ 2
dξ 2
dWi
(ξ i ) = dWi +1 (ξ i )
dξ
dξ
d 3Wi
d 3Wi +1
(
)
(ξ i ),
ξ
=
i
dξ 3
dξ 3
(14)
i = 1,..., n − 1
Substituting functions (12) into boundary and continuity conditions given by equations (13) and (14), we obtain the free vibration problem in the form
A(ω ) ⋅ X = 0
(15)
T
where X = [ A1 , B1 , C1 , D1 ,..., An , Bn , Cn , Dn ] and A(ω ) = [akj ]4 n×4 n . Two first rows of
the matrix A represent the boundary conditions at ξ = 0 and two last rows of A
represent the boundary conditions at ξ = 1 . The rows of the matrix A with nonzero elements a4i −1, 4i −3 ,…, a4i −1, 4i + 4 , a4i , 4i −3 ,…, a4i , 4i + 4 , a4i +1, 4i −3 ,…, a4i +1, 4i + 4 ,
a4i + 2, 4i −3 ,…, a4i + 2, 4i + 4 , i = 1,..., n − 1 , are determined by the continuity conditions
at ξ = ξ i , i = 1,..., n − 1 . The determinant of the matrix A has to vanish for a nontrivial solution of the frequency equation of the beam under consideration to exist.
Frequency equation
det A(ω ) = 0
(16)
is then solved numerically using an approximate method.
3. Numerical example
The numerical computations were performed for pinned-pinned FG beam
with n segments of the same length. The function f (⋅) occurring in equations
(2), (3) is assumed in one of the forms: f (ξ ) = 1 + ξ p for p = 0.5; 1.0; 2.0,
or f (ξ ) = 1+ 0.5 sin (aπξ ) for a = 1.0; 2.0; 3.0, where ξ = x L . The calculations
were carried out for various numbers of segments n = 2; 5; 10; 15.
43
Free vibration of axially functionally graded Euler-Bernoulli beams
Table 1
Four dimensionless eigenfrequencies of the FG beam with n segments of constant
length with f (ξ ) = 1 + ξ p
n
p
0.5
1
2
2
5
10
15
3.001006
3.010238
3.036439
3.045427
6.285830
6.173596
6.187237
6.198689
9.388662
9.354648
9.335179
9.344616
12.570167
12.538667
12.486328
12.489851
3.095507
3.096357
3.096695
3.096761
6.286651
6.266571
6.266667
6.266697
9.415975
9.415099
9.415025
9.415040
12.569686
12.560122
12.559585
12.559592
3.207782
3.207477
3.207418
3.207406
6.286086
6.326810
6.327096
6.327154
9.454665
9.455613
9.455755
9.455796
12.569836
12.589830
12.590206
12.590236
Table 2
Four dimensionless eigenfrequencies of the FG beam with n segments of constant
length with f (ξ ) = 1 + 0.5sin(aπξ )
n
a
1
2
3
2
5
10
15
2.810172
2.836856
2.838765
2.839073
6.278107
6.145092
6.145823
6.145838
9.339187
9.337856
9.339502
9.339522
12.568765
12.501314
12.504811
12.504871
3.141592
2.654107
2.602609
2.593108
6.283185
6.393857
6.475257
6.491527
9.424777
9.451843
9.422380
9.427523
12.566370
12.575262
12.555303
12.553347
3.460695
2.806813
2.699744
2.680683
6.268350
5.128893
4.988445
4.948780
9.573864
9.012672
9.379156
9.428255
12.573379
12.668699
12.271558
12.255414
Four dimensionless eigenfrequencies for the function f (ξ ) =1 + ξ p are presented in Table 1 and for the function f (ξ ) = 1+ 0.5 sin (aπξ ) are shown in Table 2.
44
S. Kukla, J. Rychlewska
From Table 1 it is seen that the obtained results for n = 5 and n = 10 differ at most
by 0.87% and for n = 10 and n = 15 they differ at most by 0.29%. Analogous
comparison from Table 2 gives the differences of the results 4.07 and 0.79%,
respectively.
Conclusions
The main conclusion of this contribution is that the proposed approach can be
applied to the analysis of free vibration problems for axially functionally graded
beams. The primary idea presented here is to approximate the FG beam by the
beam with piecewise exponentially varying geometrical and material properties.
The example shows that the accuracy of the numerically obtained eigenfrequencies
improves as the number of segments increases. The numerical example presented
in this paper concerns a pinned-pinned beam, but the proposed method may be
used in the vibration analysis of FG beams with other boundary conditions.
References
[1] Alshorbagy A.E., Eltaher M.A., Mahmoud F.F., Free vibration characteristics of a functionally
graded beam by finite element method, Appl. Math. Model. 2011, 35, 412-425.
[2] Hein H., Feklistova L., Free vibrations of non-uniform and axially functionally graded beams
using Haar wavelets, Eng. Struct. 2011, 33, 3696-3701.
[3] Anandrao K.S., Gupta R.K., Ramachandran P., Venkateswara Rao G., Free vibration analysis
of functionally graded beams, Def. Sci. J. 2012, 62(3), 139-146.
[4] Shahba A., Rajasekaran S., Free vibration and stability of tapered Euler-Bernoulli beams made
of axially functionally graded materials, Appl. Math. Model. 2012, 36, 3094-3111.
[5] Li X.-F., Kang Y.-A., Wu J.-X., Exact frequency equations of free vibration of exponentially
functionally graded beams, Appl. Acoustic 2013, 74, 413-420.
[6] Kukla S., Rychlewska J., Free vibration analysis of functionally graded beams, J. Appl. Math.
Comp. Mech. 2013, 12(2), 39-44.